An approach to construct the relationship between the nonlinear normal mode and forced response of nonlinear systems

An approach for correlating the given forced response with the nonlinear normal mode utilizing the modal assurance criterion is explored. The problem is transformed into a nonlinear optimization problem with nonlinear constraints. The modal assurance criterion of the Fourier coefficient vectors which indicates the degree of correlation between the nonlinear normal mode and forced response is set as the objective function whereas the nonlinear constraints are built utilizing the harmonic balance method and the Floquet theory. With the aim to make the modal assurance criterion approach to 1, the unknown optimization variables including the Fourier coefficients and the nonlinear frequency are iteratively sought using the sequential quadratic programming algorithm to fit the given periodic solution. Finally, the validity of the proposed method is demonstrated via two numerical case studies. It is illustrated that the proposed approach can be used to construct the relationship between the nonlinear normal mode and forced response. In addition, numerical examples also confirm that resonance forced responses are in the neighborhood of nonlinear normal modes.